On a boundary value problem for a system of differential equations modeling the electrical activity of the brain

A. S. Patrina

Vestnik rossijskih universitetov. Matematika, Tome 28 (2023) no. 144, pp. 383-394

Résumé

The Hopfield-type model of the dynamics of the electrical activity of the brain which is a system of differential equations of the form \begin{equation*} \dot{v}_{i}= -\alpha v_{i}+\sum_{j=1}^{n}w_{ji}f_{\delta}(v_{j})+I_{i}( t), \quad i=\overline{1,n}, \quad t\geq 0, \end{equation*} is under discussion. The model parameters are assumed to be given: $\alpha>0,$ $\tau_{ii}=0,$ $w_{ii}= 0,$ $\tau_{ji}\geq 0$ and $w_{ji}>0$ at $i\neq j,$ $I_{i}(t)\geq 0$ at $t\geq 0.$ Activation function $f_{\delta}$ ($\delta$ — the time of the transition of a neuron to the state of activity) is considered of two types: $$ \delta=0 \ \Rightarrow f_{0}(v)=\left\{ \begin{array}{ll} 0, \leq\theta,\\ 1, >\theta; \end{array}\right. \ \ \ \ \ \ \delta> 0 \ \Rightarrow \ f_{\delta}(v)=\left\{ \begin{array}{ll} 0, v\leq \theta,\\ {\delta}^{-1}( v-\theta), \theta v \leq \theta+\delta,\\ 1, >\theta+\delta. \end{array}\right.$$ For the system of differential equations under consideration, a boundary value problem with the conditions ${v_{i}(0)-v_{i}(T)=\gamma_{i},}$ $i=\overline{1,n},$ is studied. In both cases $\delta= 0$ (discontinuous function $f_{0}$) and $\delta > 0$ ($f_{0}$ continuous function), a solution exists, and if $${\delta} > \frac{T|W|_{\mathbb{R}^{n}\to \mathbb{R}^{n}}}{1 - e^{-\alpha T}}, \quad \text{where} \quad W=(w_{ij})_{n\times n}, $$ the problem has a unique solution. The work also provides estimates for the solution and its derivative. Theorems on fixed points of continuous mappings of metric and normed spaces and on fixed points of monotonic mappings of partially ordered spaces are used. The results obtained are applied to the study of periodic solutions of the differential system under consideration.

Détail

Keywords: neural network, differential equation with a discontinuous right-hand side, boundary value problem, Green's function, mappings of partially ordered spaces, periodic solution
Mots-clés : existence of a solution